Question

given: ∠5 ≅ ∠4
prove: r || s
statement
1 ∠5 ≅ ∠4
2 ∠3 ≅ ∠4
3 r || s
reason
given
given
reason?
validate
send to proof

Explanation:

Step1: Identify Given Information

We are given that $\angle 5 \cong \angle 4$. Also, we can observe that $\angle 3$ and $\angle 4$ are vertical angles (or we can use the property of alternate interior angles or corresponding angles depending on the diagram). Wait, actually, let's re - examine. If we want to prove $r\parallel s$, we can use the converse of the alternate interior angles theorem.

First, let's list the statements:

1. $\angle 5 \cong \angle 4$ (Given)
2. $\angle 3 \cong \angle 4$ (Vertical Angles Theorem: Vertical angles are congruent)
3. From 1 and 2, by the Transitive Property of Congruence, $\angle 5 \cong \angle 3$. But actually, a more direct way: If we consider $\angle 5$ and $\angle 4$ as alternate interior angles (assuming the lines and transversal are set up that way), then by the Converse of the Alternate Interior Angles Theorem, if alternate interior angles are congruent, then the lines are parallel.

Wait, let's correct the step - by - step:

Step1: Recall the Converse of Alternate Interior Angles Theorem

The Converse of the Alternate Interior Angles Theorem states that if two lines are cut by a transversal and the alternate interior angles are congruent, then the two lines are parallel.

Step2: Identify Angles as Alternate Interior Angles

In the diagram, when lines $r$ and $s$ are cut by a transversal (let's say the line that creates $\angle 4$ and $\angle 5$), $\angle 4$ and $\angle 5$ are alternate interior angles.

Step3: Use Given and Theorem to Prove Parallelism

We are given that $\angle 5 \cong \angle 4$. By the Converse of the Alternate Interior Angles Theorem, since alternate interior angles $\angle 4$ and $\angle 5$ are congruent, we can conclude that $r\parallel s$.

For the table:

- Statement 1: $\angle 5 \cong \angle 4$; Reason: Given
- Statement 2: $\angle 3 \cong \angle 4$ (Wait, no, maybe a better approach. Let's assume the transversal is the line intersecting $r$ and $s$, and $\angle 4$ and $\angle 5$ are alternate interior angles. Then:

Statement 1: $\angle 5 \cong \angle 4$; Reason: Given

Statement 2: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel (Converse of Alternate Interior Angles Theorem); Reason: Theorem

Statement 3: $r\parallel s$; Reason: From Statement 1 and Statement 2 (by applying the theorem)

But looking at the given table in the problem:

Statement 1: $\angle 5 \cong \angle 4$; Reason: Given

Statement 2: $\angle 3 \cong \angle 4$; Reason: Given (Wait, maybe $\angle 3$ and $\angle 5$ have a relation? No, maybe the vertical angles. Wait, $\angle 3$ and $\angle 4$ are vertical angles, so $\angle 3 \cong \angle 4$ by Vertical Angles Theorem. Then $\angle 5 \cong \angle 4$ (given) and $\angle 3 \cong \angle 4$ (vertical angles), so $\angle 5 \cong \angle 3$ (Transitive Property). Then $\angle 5$ and $\angle 3$ are corresponding angles (if the transversal is set up that way), so by Converse of Corresponding Angles Theorem, $r\parallel s$.

Let's re - structure:

Step1: Vertical Angles are Congruent

$\angle 3$ and $\angle 4$ are vertical angles, so $\angle 3 \cong \angle 4$ (Vertical Angles Theorem).

Step2: Use Transitive Property

We know that $\angle 5 \cong \angle 4$ (Given) and $\angle 3 \cong \angle 4$ (from Step 1). By the Transitive Property of Congruence, $\angle 5 \cong \angle 3$.

Step3: Apply Converse of Corresponding Angles Theorem

$\angle 5$ and $\angle 3$ are corresponding angles (assuming the transversal and line positions). By the Converse of the Corresponding Angles Theorem, if corresponding angles are congruent, then the lines are parallel. So $r\parallel s$.

In the given table:

- Statement 1: $\angle 5 \cong \angle 4$; Reason: Given
- Statement 2: $\angle 3 \cong \angle 4$; Reason: Vertical Angles Theorem (not Given, maybe the problem has a typo? But according to the user's table, Statement 2 has Reason "Given", maybe in the problem's context, $\angle 3 \cong \angle 4$ is also given? Then Statement 3: $r\parallel s$; Reason: If alternate interior angles (or corresponding angles) are congruent, lines are parallel.

Assuming the problem's table:

Statement 1: $\angle 5 \cong \angle 4$; Reason: Given

Statement 2: $\angle 3 \cong \angle 4$; Reason: Given

Statement 3: $r\parallel s$; Reason: If two lines are cut by a transversal and alternate interior angles (or corresponding angles) are congruent, then the lines are parallel (Converse of Alternate Interior Angles Theorem or Converse of Corresponding Angles Theorem). Since $\angle 5 \cong \angle 4$ and $\angle 3 \cong \angle 4$, then $\angle 5 \cong \angle 3$ (Transitive Property), and if $\angle 5$ and $\angle 3$ are corresponding angles, then $r\parallel s$.

The reason for Statement 3 is the Converse of the Alternate Interior Angles Theorem (or Corresponding Angles Theorem) which states that if two lines are cut by a transversal and the alternate interior (or corresponding) angles are congruent, then the lines are parallel.

Answer:

For the third statement ($r\parallel s$), the reason is the Converse of the Alternate Interior Angles Theorem (or Converse of Corresponding Angles Theorem) which states that if two lines are cut by a transversal and the alternate interior (or corresponding) angles are congruent, then the two lines are parallel. Using the given congruences $\angle 5\cong\angle 4$ and (if $\angle 3\cong\angle 4$ is used) transitive property to show congruent alternate interior/corresponding angles, we conclude $r\parallel s$.